To find the derivative of the function [tex]\( f(x) = \frac{3x^4 - 5}{x^4 + 8} \)[/tex], we can use the quotient rule of differentiation. The quotient rule states that if we have a function [tex]\( \frac{u(x)}{v(x)} \)[/tex], where both [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex] are differentiable, then its derivative is given by:
[tex]\[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \][/tex]
Here, we identify:
[tex]\[ u(x) = 3x^4 - 5 \][/tex]
[tex]\[ v(x) = x^4 + 8 \][/tex]
We need to determine the derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex]:
[tex]\[ u'(x) = \frac{d}{dx}(3x^4 - 5) = 12x^3 \][/tex]
[tex]\[ v'(x) = \frac{d}{dx}(x^4 + 8) = 4x^3 \][/tex]
Now, applying the quotient rule:
[tex]\[ f'(x) = \frac{u'v - uv'}{v^2} \][/tex]
[tex]\[ f'(x) = \frac{(12x^3)(x^4 + 8) - (3x^4 - 5)(4x^3)}{(x^4 + 8)^2} \][/tex]
First, let's simplify the numerator:
[tex]\[ (12x^3)(x^4 + 8) = 12x^7 + 96x^3 \][/tex]
[tex]\[ (3x^4 - 5)(4x^3) = 12x^7 - 20x^3 \][/tex]
Substitute these into our derivative expression:
[tex]\[ f'(x) = \frac{12x^7 + 96x^3 - (12x^7 - 20x^3)}{(x^4 + 8)^2} \][/tex]
[tex]\[ f'(x) = \frac{12x^7 + 96x^3 - 12x^7 + 20x^3}{(x^4 + 8)^2} \][/tex]
[tex]\[ f'(x) = \frac{96x^3 + 20x^3}{(x^4 + 8)^2} \][/tex]
[tex]\[ f'(x) = \frac{116x^3}{(x^4 + 8)^2} \][/tex]
Thus, the simplified form of the derivative is:
[tex]\[ f'(x) = \frac{116x^3}{(x^4 + 8)^2} \][/tex]
Upon comparing this with the provided options, we find that the correct choice is:
c.) [tex]\( f^{\prime}(x)=\frac{116 x^3}{\left(x^4+8\right)^2} \)[/tex]
